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\title{Johnston--Yang's surviving global VK psi decay improves to 0.2043325}
\author{Kenan}
\date{2026-07-19}

\begin{document}
\maketitle

\begin{abstract}
The global explicit bound for the Chebyshev function keeps Johnston--Yang's amplitude $0.026$, logarithmic power $1.801$, and full range $x\ge23$, while increasing the Vinogradov--Korobov decay constant to $0.2043325$. The proof uses only Yang's surviving zero-free denominator $51.34$, interiorized to $51.3401$, then reoptimizes the large-$x$ tail and closes the remaining ranges with finite, Buethe, and C-0067 splices.
\end{abstract}

\[
|\psi(x)-x|<0.026x(\log x)^{1.801}\exp\!\left(-0.2043325\frac{(\log x)^{3/5}}{(\log\log x)^{1/5}}\right)\quad(x\ge23)
\]

\section{The improved envelope}

For $L=\log x$ and $r(L)=L^{3/5}/(\log L)^{1/5}$, the claimed envelope holds for every real $x\ge23$. It improves the published decay $0.1853$ and the previous certified same-form decay $0.2043$ without changing any other headline parameter.

Because the amplitude, power, and domain are identical, the larger decay constant makes the new right-hand side strictly smaller at every point of the common range.

\[
|\psi(x)-x|<0.026xL^{1.801}e^{-0.2043325r(L)}
\]

\[
0.2043325-0.1853=\frac{7613}{400000},\qquad 0.2043325-0.2043=\frac{13}{400000}
\]
\section{The surviving zero-free input}

Yang's theorem gives an open Vinogradov--Korobov zero-free region with denominator $51.34$. Replacing it by the closed denominator $c_*=51.3401$ moves the boundary strictly inside that open region and introduces an exact padding of $10^{-4}$.

Minimizing the Johnston--Yang height expression then gives a lower coefficient $Q(c_*)=0.2043529808\ldots$, which is safely above the target decay. No input from the retracted $51.323$ branch is used.

\[
\zeta(\sigma+it)\ne0\quad\left(\sigma>1-\frac1{51.34(\log t)^{2/3}(\log\log t)^{1/3}}\right)
\]

\[
Q(c)=\left(\frac5{3c^3}\right)^{1/5}\left[\left(\frac32\right)^{2/5}+\left(\frac23\right)^{3/5}\right]
\]
\section{Large-x tail}

The proof chooses $L_0=950000000$, $\sigma=0.9999714$, $\theta=0.08917$, and $B_3=0.21617$. Directed interpolation of the source density rows supplies the associated constants, while the fixed trial proves $Q(c_*)r(L)\le\log T\le B_3r(L)$ for every $L\ge L_0$.

The repaired explicit-formula terms are normalized by the target envelope. Each normalized contribution is proved decreasing from $L_0$ onward, and their endpoint sum is below one with reserve about $0.0501$. The native first-density decay is $0.2043373955\ldots$, still strictly above $0.2043325$.

\[
\frac{s_1+s_2+s_3}{0.026L^{1.801}e^{-0.2043325r(L)}}\le0.9498732528858\ldots<1\qquad(L\ge L_0)
\]

\begin{enumerate}[leftmargin=*]
\item Interiorize the open zero-free denominator and derive the coefficient $Q(c_*)$.
\item Choose a source-admissible density tuple and prove the upper and lower height bounds uniformly from $L_0$.
\item Normalize every explicit-formula term by the target envelope and reduce the tail to endpoint inequalities plus monotonicity.
\end{enumerate}
\section{Closing the full range}

For $23\le x\le59$, the verifier uses that $\psi(x)$ is constant between prime powers and checks both endpoints of every interval. From $59$ to $e^{58}$ it compares against Johnston--Yang's explicit Buethe bound.

For $58\le L\le L_0$, the target is compared with C-0067's global FKS--Yang bound. An analytic derivative reduction shows that the logarithmic comparison has no interior minimum, so positive endpoint margins cover the entire middle interval.

\[
\frac{|\psi(x)-x|}{x}<9.2202181L^{3/2}e^{-0.88178\sqrt L}
\]
\section{What the certificate establishes}

The directed 256-bit Arb verifier hash-pins the Johnston--Yang source, Yang's thesis, C-0067, C-0070, and a forward use of the published bound. It checks the source interpolation, the open-to-closed padding, the exact $Q(c)$ calculation, every tail derivative, every range splice, and both exact improvements.

It also reruns the repaired published $0.1853$ theorem, including the printed $B_2$ defect, so the comparison baseline is reproduced rather than assumed. A separately pinned independent checker is corroborative; the interval bounds and analytic monotonicity reductions are the proof.

\section{Pinned certificate}
The public verifier certifies the source pins, the surviving zero-free input, the reoptimized large-x tail, all finite and middle-range splices, and the exact comparison with both prior same-form bounds. The independent checker is an additional corroborative artifact, not the logical basis of the theorem.

\subsection*{Run command}
\begin{verbatim}
uv run --frozen python canon/witnesses/C-0099/verify.py
\end{verbatim}

\subsection*{Pinned files}
\begin{itemize}[leftmargin=*]
\item \path{canon/witnesses/C-0099/verify.py}
\item \path{canon/witnesses/C-0099/independent_check.py}
\item \path{canon/witnesses/C-0099/PROOF.md}
\item \path{canon/witnesses/C-0099/PIN.md}
\end{itemize}

\section{Scope}
The global bound keeps amplitude $0.026$, logarithmic power $1.801$, and range $x\ge23$, using the denominator $51.34$ closed inward to $51.3401$.

\section{Sources}
\begin{itemize}[leftmargin=*]
\item Canonical claim: \path{canon/claims/C-0099-johnston-yang-surviving-global-vk-psi-decay-0-2043325.md}
\item Proof: \path{canon/witnesses/C-0099/PROOF.md}
\item Source and verifier pin: \path{canon/witnesses/C-0099/PIN.md}
\end{itemize}

\end{document}
